A rod of metal X of length 50.0 cm elongates by 0.10 cm when it is heated from `0^(@)C" to "100^(@)C`. Another rod o metal Y of length 80.0 cm elongates by 0.08 cm for the same rise in temperature, A third rod of length 50 cm made by welding pieces of rods X and Y placed end to end, elongates by 0.03 cm when its temperature is raised from `0^(@)C" to "50^(@)C`.
The length of the rod of metal X in the composite piece is

Options

A. 10 cm
B. 20 cm
C. 30 cm
D. 40 cm

Answer

A

Explanation

In the composite rod, `L_(x) +L_(y)=50.0` cm. When this rod is heated by `50^(@)C`, let the new lengths be `L_(x). and L_(y).`. Given `L_(x).+L_(y).=50.0+0.03=50.03` cm.
Here `L_(x). =L_(x) (1+alpha_(x) xx50) and L_(y). =L_(y) (1+ alpha_(y) xx 50)`.
Therefore, `50.03 = L_(x) + L_(y) +(L_(x) alpha_(x) + L_(y) alpha_(y)) 50=50 +(L_(x) alpha_(x) + L_(y) alpha_(y)) 50`
Substituting the values of `a_(x) and a_(y)` and noting that `L_(x) + L_(y) =50` cm, we get `L_(x) =10` cm.